Smarandache Curves in Minkowski Space-time

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1 International J.Math. Combin. Vol.3 (2008), 5-55 Smarandache Curves in Minkowski Space-time Melih Turgut and Süha Yilmaz (Department of Mathematics of Buca Educational Faculty of Dokuz Eylül University, 3560 Buca-Izmir,Turkey.) Abstract: A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache Curve. In this paper, we define a special case of such curves and call it Smarandache TB 2 Curves in the space E 4. Moreover, we compute formulas of its Frenet apparatus according to base curve via the method expressed in [3]. By this way, we obtain an another orthonormal frame of E 4. Key Words: Minkowski space-time, Smarandache curves, Frenet apparatus of the curves. AMS(2000): 53C50, 5B20.. Introduction In the case of a differentiable curve, at each point a tetrad of mutually orthogonal unit vectors (called tangent, normal, first binormal and second binormal) was defined and constructed, and the rates of change of these vectors along the curve define the curvatures of the curve in Minkowski space-time []. It is well-known that the set whose elements are frame vectors and curvatures of a curve, is called Frenet Apparatus of the curves. The corresponding Frenet s equations for an arbitrary curve in the Minkowski space-time E 4 are given in [2]. A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache Curve. We deal with a special Smarandache curves which is defined by the tangent and second binormal vector fields. We call such curves as Smarandache TB 2 Curves. Additionally, we compute formulas of this kind curves by the method expressed in [3]. We hope these results will be helpful to mathematicians who are specialized on mathematical modeling. 2. Preliminary notes To meet the requirements in the next sections, here, the basic elements of the theory of curves in the space E 4 are briefly presented. A more complete elementary treatment can be found in the reference []. Minkowski space-time E 4 is an Euclidean space E4 provided with the standard flat metric given by Received August 6, Accepted September 2, 2008.

2 52 Melih Turgut and Suha Yilmaz g = dx 2 + dx2 2 + dx2 3 + dx2 4, where (x, x 2, x 3, x 4 ) is a rectangular coordinate system in E 4. Since g is an indefinite metric, recall that a vector v E 4 can have one of the three causal characters; it can be space-like if g(v, v) > 0 or v = 0, time-like if g(v, v) < 0 and null (light-like) if g(v, v)=0 and v 0. Similarly, an arbitrary curve α = α(s) in E 4 can be locally be space-like, time-like or null (light-like), if all of its velocity vectors α (s) are respectively space-like, time-like or null. Also, recall the norm of a vector v is given by v = g(v, v). Therefore, v is a unit vector if g(v, v) = ±. Next, vectors v, w in E 4 are said to be orthogonal if g(v, w) = 0. The velocity of the curve α(s) is given by α (s). Denote by {T(s), N(s), B (s), B 2 (s)} the moving Frenet frame along the curve α(s) in the space E 4. Then T, N, B, B 2 are, respectively, the tangent, the principal normal, the first binormal and the second binormal vector fields. Space-like or time-like curve α(s) is said to be parametrized by arclength function s, if g(α (s), α (s)) = ±. Let α(s) be a curve in the space-time E 4, parametrized by arclength function s. Then for the unit speed space-like curve α with non-null frame vectors the following Frenet equations are given in [2]: T N B B 2 = 0 κ 0 0 κ 0 τ 0 0 τ 0 σ 0 0 σ 0 T N B B 2, () where T, N, B and B 2 are mutually orthogonal vectors satisfying equations g(t, T) = g(n, N) = g(b, B ) =, g(b 2, B 2 ) =. Here κ, τ and σ are, respectively, first, second and third curvature of the space-like curve α. In the same space, in [3] authors defined a vector product and gave a method to establish the Frenet frame for an arbitrary curve by following definition and theorem. Definition 2. Let a = (a, a 2, a 3, a 4 ), b = (b, b 2, b 3, b 4 ) and c = (c, c 2, c 3, c 4 ) be vectors in E 4. The vector product in Minkowski space-time E4 is defined by the determinant e e 2 e 3 e 4 a a 2 a 3 a 4 a b c =, (2) b b 2 b 3 b 4 c c 2 c 3 c 4 where e, e 2, e 3 and e 4 are mutually orthogonal vectors (coordinate direction vectors) satisfying equations e e 2 e 3 = e 4, e 2 e 3 e 4 = e, e 3 e 4 e = e 2, e 4 e e 2 = e 3.

3 Smarandache Curves in Minkowski Space-time 53 Theorem 2.2 Let α = α(t) be an arbitrary space-like curve in Minkowski space-time E 4 with above Frenet equations. The Frenet apparatus of α can be written as follows; T = α α, (3) N = α 2.α g(α, α ).α α 2, (4).α g(α, α ).α B = µn T B 2, (5) and κ = B 2 = µ T N α T N α, (6) α 2.α g(α, α ).α α 4 (7) T N α. α τ = α 2 (8).α g(α, α ).α σ = g(α (IV ), B 2 ) T N α. α, (9) where µ is taken or + to make + the determinant of [T, N, B, B 2 ] matrix. 3. Smarandache Curves in Minkowski Space-time Definition 3. A regular curve in E 4, whose position vector is obtained by Frenet frame vectors on another regular curve, is called a Smarandache Curve. Remark 3.2 Formulas of all Smarandache curves Frenet apparatus can be determined by the expressed method. Now, let us define a special form of Definition 3.. Definition 3.3 Let ξ = ξ(s) be an unit space-like curve with constant and nonzero curvatures κ, τ and σ; and {T, N, B, B 2 } be moving frame on it. Smarandache TB 2 curves are defined with X = X(s X ) = κ2 (s) + σ 2 (s) (T(s) + B 2(s)). (0) Theorem 3.4 Let ξ = ξ(s) be an unit speed space-like curve with constant and nonzero curvatures κ, τ and σ and X = X(s X ) be a Smarandache TB 2 curve defined by frame vectors of ξ = ξ(s). Then

4 54 Melih Turgut and Suha Yilmaz (i) The curve X = X(s X ) is a space-like curve. (ii) Frenet apparatus of {T X, N X, B X, B 2X, κ X, τ X, σ X } Smarandache TB 2 curve X = X(s X ) can be formed by Frenet apparatus {T, N, B, B 2, κ, τ, σ} of ξ = ξ(s). Proof Let X = X(s X ) be a Smarandache TB 2 curve defined with above statement. Differentiating both sides of (0), we easily have The inner product g(x, X ) follows that dx ds X ds X ds = κ2 (s) + σ 2 (s) (κn + σb ). () g(x, X ) =, (2) where denotes derivative according to s. (2) implies that X = X(s X ) is a space-like curve. Thus, the tangent vector is obtained as T X = κ2 (s) + σ 2 (s) (κn + σb ). (3) Then considering Theorem 2., we calculate following derivatives according to s: Then, we form X = X = κ2 + σ 2 ( κ2 T τσn + κτb + σ 2 B 2 ). (4) κ2 + σ 2 [κτσt + ( κ3 κτ 2 )N + (σ 3 τ 2 σ)b + κτσb 2 ]. (5) X (IV ) = κ2 + σ 2 [(...)T + (...)N + (...)B + (σ 4 τ 2 σ 2 )B 2 ]. (6) X 2.X g(x, X ).X = Equation (7) yields the principal normal of X as κ2 + σ 2 [ κ2 T τσn + κτb + σb 2 ]. (7) N X = κ2 T τσn + κτb + σb 2 κ4 + τ 2 σ 2 + κ 2 τ 2 + σ. (8) 2 Thereafter, by means of (7) and its norm, we write first curvature κ4 + τ κ X = 2 σ 2 + κ 2 τ 2 + σ 2 κ 2 + σ 2. (9) The vector product T X N X X follows that T X N X X = [κσ(κ 2 + σ 2 )(τ 2 σ)t + κτσ 2 (κ 2 + σ)n A κ 2 τσ(κ 2 + σ)b + κτ(κ 2 + σ 2 )(κ 2 + τ 2 )B 2 ], (20) where, A =. Shortly, let us denote T ( κ 4 +τ 2 σ 2 +κ 2 τ 2 +σ 2 )(κ 2 +σ 2 X N X X with l T + ) l 2 N + l 3 B + l 4 B 2. And therefore, we have the second binormal vector of X = X(s X ) as B 2X = µ l T + l 2 N + l 3 B + l 4 B 2 l 2 + l2 2 + l (2) l2 4

5 Smarandache Curves in Minkowski Space-time 55 Thus, we easily have the second and third curvatures as follows: ( l 2 τ X = + l2 2 + l2 3 + l2 4 )(κ2 + σ 2 ) κ 4 + τ 2 σ 2 + κ 2 τ 2 + σ 2, (22) σ 2 (σ 2 τ 2 ) σ X = (κ 2 + σ 2 ) l 2 + l2 2 + l (23) l2 4 Finally, the vector product N X T X B 2X gives us the first binormal vector B X = µ [(κσl 3 σ 2 l 2 τ(κ 2 + σ 2 )l 4 ]T σ(κ 2 l 4 + σl )N L +κ(κ 2 l 4 + σl )B + [κ 2 (σl 2 κ 2 l 3 ) + τl (κ 2 + σ 2 )]B 2, (24) where L =. ( l 2 +l 2 2 +l2 3 +l2 4 )(κ2 +σ 2 )( κ 4 +τ 2 σ 2 +κ 2 τ 2 +σ 2 ) Thus, we compute Frenet apparatus of Smarandache TB 2 curves. Corollary 3. Suffice it to say that {T X, N X, B X, B 2X } is an orthonormal frame of E 4. Acknowledgement The first author would like to thank TUBITAK-BIDEB for their financial supports during his Ph.D. studies. References [] B. O Neill, Semi-Riemannian Geometry, Academic Press, New York, 983. [2] J. Walrave, Curves and surfaces in Minkowski space. Dissertation, K. U. Leuven, Fac. of Science, Leuven, 995. [3] S. Yilmaz and M. Turgut, On the Differential Geometry of the curves in Minkowski spacetime I, Int. J. Contemp. Math. Sci. 3(27), , 2008.